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Your proto-Euclidean algorithm is basically equivalent to the Greedy algorithm for finding the alternating Egyptian fraction representation of a rational. For instance, in your example, if we expand the nested parentheses we get: $$\frac{22}{51} = \frac{1}{2}(1-\frac{1}{7}(1-\frac{1}{25}(1-\frac{1}{51}))) = \frac{1}{2}-\frac{1}{14}+\frac{1}{350}-\frac{1}{17850}$$